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We propose a correspondence between vertex operator superalgebras and families of sigma models in which the two structures are related by symmetry properties and a certain reflection procedure. The existence of such a correspondence is motivated by previous work on N=(4,4) supersymmetric non-linear sigma models on K3 surfaces and on a vertex operator superalgebra with Conway group symmetry. Here we present an example of the correspondence for N=(4,4) supersymmetric non-linear sigma models on four-tori, and compare it to the K3 case., 31 pages including three appendices

Umbral moonshine connects the symmetry groups of the 23 Niemeier lattices with 23 sets of distinguished mock modular forms. The 23 cases of umbral moonshine have a uniform relation to symmetries of K3 string theories. Moreover, a supersymmetric vertex operator algebra with Conway sporadic symmetry also enjoys a close relation to the K3 elliptic genus. Inspired by the above two relations between moonshine and K3 string theory, we construct a chiral CFT by orbifolding the free theory of 24 chiral fermions and two pairs of fermionic and bosonic ghosts. In this paper we mainly focus on the case of umbral moonshine corresponding to the Niemeier lattice with root system given by 6 copies of D4 root system. This CFT then leads to the construction of an infinite-dimensional graded module for the umbral group $${G^{D_4^{\oplus 6}}}$$ whose graded characters coincide with the umbral moonshine functions. We also comment on how one can recover all umbral moonshine functions corresponding to the Niemeier root systems $${A_5^{\oplus 4}D_4}$$ , $${A_7^{\oplus 2}D_5^{\oplus 2}}$$ , $${A_{11}D_7 E_6}$$ , $${A_{17}E_7}$$ , and $${D_{10}E_7^{\oplus 2}}$$ .

We introduce a machine learning model, the q-CNN model, sharing key features with convolutional neural networks and admitting a tensor network description. As examples, we apply q-CNN to the MNIST and Fashion MNIST classification tasks. We explain how the network associates a quantum state to each classification label, and study the entanglement structure of these network states. In both our experiments on the MNIST and Fashion-MNIST datasets, we observe a distinct increase in both the left/right as well as the up/down bipartition entanglement entropy (EE) during training as the network learns the fine features of the data. More generally, we observe a universal negative correlation between the value of the EE and the value of the cost function, suggesting that the network needs to learn the entanglement structure in order the perform the task accurately. This supports the possibility of exploiting the entanglement structure as a guide to design the machine learning algorithm suitable for given tasks.

The word moonshine refers to unexpected relations between the two distinct mathematical structures: finite group representations and modular objects. It is believed that the key to understanding moonshine is through physical theories with special symmetries. Recent years have seen a varieties of new ways in which finite group representations and modular objects can be connected to each other, and these developments have brought promises and also puzzles into the string theory community. These lecture notes aim to bring graduate students in theoretical physics and mathematical physics to the forefront of this active research area. In Part II of this note, we review the various cases of moonshine connections, ranging from the classical monstrous moonshine established in the last century to the most recent findings. In Part III, we discuss the relation between the moonshine connections and physics, especially string theory. After briefly reviewing a recent physical realisation of monstrous moonshine, we will describe in some details the mystery of the physical aspects of umbral moonshine, and also mention some other setups where string theory black holes can be connected to moonshine. To make the exposition self-contained, we also provide the relevant background knowledge in Part I, including sections on finite groups, modular objects, and two-dimensional conformal field theories. This part occupies half of the pages of this set of notes and can be skipped by readers who are already familiar with the relevant concepts and techniques., summer school lecture notes, 78 pages